Non-preservation of $α$-concavity for the porous medium equation

Abstract

We show that the porous medium equation does not in general preserve $α$-concavity of the pressure for $0\leα<1/2$ or $1/2<α\le 1$. In particular, this resolves an open problem of Vázquez on whether concavity of pressure is preserved by the porous medium equation. Our results strengthen an earlier work of Ishige-Salani, who considered the case of small $α>0$. Since Daskalopoulos-Hamilton-Lee showed that $1/2$-concavity is preserved, our result is sharp. Our explicit examples show that concavity can be instantaneously broken at an interior point of the support of the initial data. For $0\leα<1/2$, we give another set of examples to show that concavity can be broken at a boundary point.

Figures (1)

• Figure 1:

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• Lemma 3.1
• proof
• proof : Proof of Theorem \ref{['maintheorem0']}
• Lemma 4.1
• proof
• Lemma 4.2